suppose celine wants to choose a box that maximizes the amount of cereal it can hold.\nvolume of box 1…

suppose celine wants to choose a box that maximizes the amount of cereal it can hold.\nvolume of box 1: $3x^{5}$\nvolume of box 2: $4x^{5}-x^{4}$\nif celine decides the width of the cereal boxes will be greater than 1, which box will hold more cereal? explain.

suppose celine wants to choose a box that maximizes the amount of cereal it can hold.\nvolume of box 1: $3x^{5}$\nvolume of box 2: $4x^{5}-x^{4}$\nif celine decides the width of the cereal boxes will be greater than 1, which box will hold more cereal? explain.

Answer

Explanation:

Step1: Find the difference of volumes

Let (V_1 = 3x^{5}) and (V_2=4x^{5}-x^{4}). Calculate (V_2 - V_1=(4x^{5}-x^{4})-3x^{5}). [ \begin{align*} V_2 - V_1&=4x^{5}-x^{4}-3x^{5}\ &=x^{5}-x^{4}\ &=x^{4}(x - 1) \end{align*} ]

Step2: Analyze the sign of the difference

We know that (x>1). Since (x>1), then (x - 1>0) and (x^{4}>0) for (x>1). So (V_2 - V_1=x^{4}(x - 1)>0) when (x > 1).

Answer:

Box 2 will hold more cereal. When (x>1), the difference (V_2 - V_1=x^{4}(x - 1)>0), which means (V_2>V_1).